For a system of quadratic differential equations, such as (REF ), a natural discretization has
been constructed by Kahan [1]}, leading to a rational map, called its Kahan map (see also
[2]}, [3]}). Applied to (REF ), the Kahan map with step size \(\varepsilon \in \mathbb {F}^*\)  is the
rational map, corresponding to the field automorphism  \(\widetilde{}\;:\mathbb {F}(x_1,\dots ,x_n)\rightarrow \mathbb {F}(x_1,\dots ,x_n)\) , defined by the following formulas, where the right hand side has been obtained from
(REF ) by polarization:
\(\frac{\tilde{x}_i -x_i}{\varepsilon } = \sum _{j=1}^n a_{i,j} \left(\tilde{x}_i x_j+x_i\tilde{x}_j\right)\;, \ \ i=1,2, \dots , n \; .\) 
